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COMPRESSIBLE FLOW

COMPRESSIBLE FLOW Introduction The compressibility of a fluid is, basically, a measure of the change in density that will be produced in the fluid by a specific change in pressure and temperature. In general, gases are highly compressible and liquids have a very low compressibility.

Part one : Introduction of Compressible Flow 1

COMPRESSIBLE FLOW

Application ; Aircraft design Gas and steam turbines Reciprocating engines Natural gas transmission lines Combustion chambers Compressibility effect ; Supersonic – the flow velocity is relatively high compared to the speed of sound in the gas. Subsonic


COMPRESSIBLE FLOW

Fundamental assumptions

1. The gas is continuous. 2. The gas is perfect (obeys the perfect gas law) 3. Gravitational effects on the flow field are

negligible. 4. Magnetic and electrical effects are negligible. 5. The effects of viscosity are negligible.

Applied principles

1. Conservation of mass (continuity equation) 2. Conservation of momentum (Newton’s law) 3. Conservation of energy (first law of

thermodynamics) 4. Equation of state


COMPRESSIBLE FLOW

Perfect gas law :

RTP=

ρ

P : Pressure ρ : Density R : Universal gas constant

)(04.287 KkgJ

airR ⋅=

T : Temperature


COMPRESSIBLE FLOW

Conservation laws : Conservation of mass

Rate mass enters control volume

Rate mass leaves control volume = _

Rate of increase of mass of fluid in

control volume

Conservation of momentum :

Net force on gas in control volume

in direction considered

Rate of increase of momentum in

direction considered of

fluid in control l

Rate momentum leaves control

volume in direction

considered

Rate momentum leaves control

volume in direction

considered

= +

_


COMPRESSIBLE FLOW

Conservation of energy :

Rate of increase in internal energy

and kinetic energy of gas in control volume

Rate enthalpy and kinetic

energy leave control volume

Rate enthalpy and kinetic

energy enter control volume

+ _

Rate heat is transferred into control volume

Rate work is done by gas in control

volume = _


COMPRESSIBLE FLOW

Definition : A control volume is a volume in space (geometric entity, independent of mass) through which fluid may flow Enthalpy H, is the sum of internal energy U and the product of pressure P and volume V appears.

PVUH +=


COMPRESSIBLE FLOW

COMPRESSIBLE FLOW

Introduction

Many of the compressible flows that occur in

engineering practice can be adequately modeled as a

flow through a duct or streamtube whose cross-sectional

area is changing relatively slowly in the flow direction.

A duct is a solid walled channel, whereas a streamtube

is defined by considering a closed curve drawn in a

fluid flow.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow 8

COMPRESSIBLE FLOW

Quasi-one-dimensional flow is flows in which the flow

area is changing but in which the flow at any section

can be treated as one-dimensional.


COMPRESSIBLE FLOW

CONTINUITY EQUATION

The continuity equation is obtained by applying the

principle of conservation of mass to flow through a

control volume.

One-dimensional flow is being considered.

There is no mass transfer across the control volume.

The only mass transfer occurs through the ends of the

control volume.


COMPRESSIBLE FLOW

Mass enters through the left hand face of the control

volume be equal to the rate at which mass leaves

through the right hand face of the control volume.

21 mm && =

We know that VAm ρ=&

We considered ;

222111 AVAV ρρ =

For the differentially short control volume indicated,


COMPRESSIBLE FLOW

above equation gives ;

))()(( dAAdVVdVA +++= ρρρ

Neglecting higher order terms, we found ;

0=++ VdAAdVVAd ρρρ

ividing this equation by VAρ D then gives ;

0=+dVd

+AdA

Vρρ

This equation relates the fractional changes in density,

velocity and area over a short length of the control

volume.


COMPRESSIBLE FLOW

MOMENTUM EQUATION (Euler’s equation)

The flow is steady flow.

Gravitational forces are being neglected.

The only forces acting on the control volume are the

pressure forces and the frictional force exerted on the

surface of the control volume.


COMPRESSIBLE FLOW

The net force on the control volume in the x-direction

is ;

dFAdAAdPPPdAAdPpPA ++++++ ]))][(([())(( 2

1

Note :

dx is too small, dPdA have been neglected.

Mean pressure on the curved surface can be taken

as the average of the pressures acting on the two end

surfaces.

dFµ is the frictional force.

Rearranging above equation, we found the net force on

the control volume in the x-direction is ;

dFAdP


COMPRESSIBLE FLOW

Since the rate at which momentum crosses any section

of the duct is equal to , we found that ; Vm&VAdVVdVVVA ρρ =+ ])[(

The above equation can be written as ;

VAdVdFAd ρρ =

Frictional force is assumed to be negligible. The Euler’s

equation for steady flow through a duct becomes;

VdVdP

=ρ


COMPRESSIBLE FLOW

Integrating Euler’s equation ;

CdPV

=+ρ2

2

(For compressible)

And if density can be assumed constant, Euler’s

equation become ;

CPV=+

ρ2

2

(For incompressible)


COMPRESSIBLE FLOW

STEADY FLOW ENERGY EQUATION

For flow through the type of control volume considered as before, we found ;

wqV

hV

h ++=+22

21

1

22

2

h = enthalpy per mass

V = velocity

q = heat transferred into the control volume per unit mass of fluid

flowing through it w = work done by the fluid per unit mass


COMPRESSIBLE FLOW

Assumption ; No work is done, w=0

Perfect gases is considered, Tch p=

Steady flow energy equation ;

qV

TcV

Tc pp ++=+22

21

1

22

2

Applying this equation to the flow through the

differentially short control volume gives ;

2)()(

2

22 dVVdTTcdq

VTc pp

+++=++


COMPRESSIBLE FLOW

Neglecting higher order terms gives ;

dqVdVdTcp =+

This equation indicates that in compressible flows, changes in velocity will, in general, induce changes in temperature and that heat addition can cause velocity changes as well as temperature changes. If the flow is adiabatic i.e., if there is no heat transfer to of from the flow, it gives ;

22

21

1

22

2V

TcV

Tc pp +=+


COMPRESSIBLE FLOW

Steady flow energy equation for adiabatic flow becomes ;

0=+VdVdTcp

This equation shows that in adiabatic flow, an increase

in velocity is always accompanied by a decrease in

temperature.


COMPRESSIBLE FLOW

EQUATION OF STATE

When applied between any two points in the flow ;

22

2

11

1

TP

TP

ρρ=

When applied between the inlet and the exit of a

differentially short control volume, this equation

becomes ;

))(( dTTddPP

TP

+++

=ρρρ


COMPRESSIBLE FLOW

Higher order terms are neglected and it gives ;

TdTd

PdP

TP

TP

+= 111 ρρ

ρρ

0=TdTd

PdP

ρρ

This equation shows how the changes in pressure,

density and temperature are interrelated in compressible

flow.


COMPRESSIBLE FLOW

ENTROPY CONSIDERATIONS

In studying compressible flows, another variable, the

entropy, s, has to be introduced. The entropy basically

places limitations on which flow processes are

physically possible and which are physically excluded.

The entropy change between any two points in the flow

is given by ;

1

2

1

2 lnln12 PP

TT

p Rcss = (1)

Since , this equation can be written; vp ccR =

=1

121

2

1

2ln PP

TT

pcss

If there is no change in entropy, i.e., if the flow is


COMPRESSIBLE FLOW

isentropic, this equation requires that : 1

1

2

1

2 =PP

TT

hence, since the perfect gas law gives ;

2

1

1

2

1

2

ρρ

PP

TT

=

it follows that in isentropic flow :

ρρ

=1

2

1

2

PP

in isentropic flows, then ρP is a constant.

If equation (1) is applied between the inlet and the exit


COMPRESSIBLE FLOW

of a differentially short control volume, it gives ;

PdPP

TdTT

p Rcsdss ++=+ lnln)(

neglecting small value, the above equation gives;

PdP

TdT

p Rcds = (2)

which can be written as ;

PdP

TdT

cds

p

=1

lastly, it is noted that in an isentropic flow, equation (2)

gives;

dPPRT

dTcp =

using the perfect gas law ;


COMPRESSIBLE FLOW

ρdP

dTcp = (3)

but the energy equation for isentropic flow, i.e., for flow

with no heat transfer, it gives ;

0=+VdVdTcp

which using equation (3) gives ;

0=+VdVdPρ


COMPRESSIBLE FLOW

SOME FUNDAMENTAL ASPECTS

OF COMPRESSIBLE FLOW

Mach number

aVM ==

sound of speed velocitygas number,mach

RTPa γργ

==

1<M : subsonic

1=M : transonic

1>M : supersonic

1>>M : hypersonic

Part three : Mach Number 27

COMPRESSIBLE FLOW

Isentropic flow in a streamtube

In order to illustrate the importance of the Mach number

in determining the conditions under which

compressibility must be taken in account, isentropic

flow, i.e., frictionless adiabatic flow, through a

streamtube will be first considered.

From previous chapter, we know that ;

VdV

PV

PdP 2ρ

−= and γρ

2aP=

the above equation can be written as :

VdVM

VdV

aV

PdP 2

2

2

γγ −=−= (1)

This equation shows that the magnitude of the fractional

pressure change, induced by a given fractional velocity

change, depends on the square of Mach number.


COMPRESSIBLE FLOW

Next, consider the energy equation. Since adiabatic

flow is being considered ;

VdVM

cR

VdV

TcV

TdT

pp

22 γ

−=−=

Since; γ11−=−= vp ccR and 1−= γγ

pcR

Above equation can be written as ;

VdVM

TdT 2)1( −−= γ (2)

Lastly, consider the equation of state;

TdTd

PdP

+=ρρ

combining above equation with eq.(1) and eq.(2)


COMPRESSIBLE FLOW

VdVM

VdVMd 22 )1( −+−= γγ

ρρ

This equation indicates that:

2MV

dV

d−=ρ

ρ

(negative sign means, density decrease when velocity

increased)

at M=0.1 , %1−=V

dV

dρ

ρ

at M=0.33 , %11−=V

dV

dρ

ρ

At low mach number, density changes will be

insignificant.


COMPRESSIBLE FLOW

Normally at M<0.3, the fluid is assumed

incompressible.

It should also be noted that above equation can we

written as ;

2)1( MVdVTdT

−−= γ

Similarly, the temperature difference is neglected at

lower value of Mach number.


COMPRESSIBLE FLOW

Mach waves

Disturbances tend to propagated ahead of the body in

motion to “warn” the gas of the approach of the body.

This is due to pressure at the surface is higher than

surrounding gas and pressure waves spread out from the

body.

The pressure waves spread out at the of sound

Effect of the velocity of the body relative to the speed

of sound (pressure wave velocity) on the flow field.


COMPRESSIBLE FLOW

Consider for subsonic flow M<1, figure (1).

Speed of the body u and speed of sound a, where u<a.

Body position at a, b, c and d at time interval t. Waves

generated at time 0, t, 2t and 3t. Since u<a, a body

moves slower than the waves and therefore a body will

never overtake it.


COMPRESSIBLE FLOW

If u>a, then M>1, the flow is supersonic, a body moves

faster than the waves and will overtake it, (figure (2)).

The waves lie within a cone which has its vertex at the

body at the time considered. On gas within this cone

“aware” of the presence of the body. Vertex angle α is

called Mach angle, where ;

Mua 1sin ==α

The cone is therefore termed a conical Mach wave.


COMPRESSIBLE FLOW


COMPRESSIBLE FLOW


COMPRESSIBLE FLOW

ONE-DIMENSIONAL

ISENTROPIC FLOW

INTRODUCTION

An adiabatic flow (a flow in which there is no heat

exchange) in which viscous losses are negligible, i.e., it

is an adiabatic frictionless flow.

Although no real flow is entirely isentropic, there are

many flows of great practical importance in which the

major portion of the flow can be assumed to be

isentropic.

Part four : One-Dimensional Isentropic Flow 37

COMPRESSIBLE FLOW

For example, in internal duct flows there are many

important cases where the effects of viscosity and heat

transfer are restricted to thin layers adjacent to the walls,

i.e., are only important in the wall boundary layers, and

the rest of the flow can be assumed to be isentropic.

Even when non-isentropic effects become important, it

is often possible to calculate the flow by assuming it to

be isentropic and to then apply an empirical correction

factor to the solution so obtained to account for the

non-isentropic effect, for example, in the design nozzle.


COMPRESSIBLE FLOW

GOVERNING EQUATION

By definition, the entropy remains constant in an

isentropic flow.

cP=γρ (c:constant) (4.1)

From equation (4.1) γ

ρρ

=1

2

1

2

PP


COMPRESSIBLE FLOW

Hence, since the general equation of state gives ;

22

2

11

1

TP

TP

ρρ= or

1

2

1

2

1

2

ρρ

PP

TT

=

It follows that in isentropic flow ;

γγγ

ρρ

1

1

2

1

1

2

1

2

−−

==PP

TT

Recalling that RTa γ= , that ;

γγγ

ρρ

ρρ 2

1

1

22

1

1

221

1

2

1

2

−−

===TT

aa

eq.(4.5)

The steady flow adiabatic energy equation is next

applied between the point 1 and point 2. This gives ;

22

22

2

21

1VTcVTc pp +=+


COMPRESSIBLE FLOW

It can be written as ;

)2(1)2(1 1

212 TcVT p+

=2

221 TcVT p+

rom ;

F

222V

=2

122

McR

RTV

Tc pp

−=γγ

γ

o, it follows that ; S

222

1

212

12 )1(1 MT −+=

γ

1 )1(1 MT −+ γ eq.(4.6)

his equation applies in adiabatic flow. If friction T

effects are also negligible, i.e., if the flow is isentropic,

eq.(4.6) cam be used in conjunction with the isentropic

state relations given in eq.(4.5) to obtain ;


COMPRESSIBLE FLOW

1

222

1

212

1

1

2

)1(1)1(1 −

−+−+

=γγ

γγ

MM

PP

and

11

222

1

212

1

1

2

)1(1)1(1 −

−+−+

=γ

γγ

ρρ

MM

astly, it is called that the continuity equation gives ; L

222111 AVAV ρρ =

hich can be rearranged to give ; w

112 VA ρ

= 221 VA ρ


COMPRESSIBLE FLOW

STAGNATION CONDITIONS

Stagnation conditions are those that would exist if the

flow at any point in fluid stream was isentropically

brought to rest.

If the entire flow is essentially isentropic and if the

velocity is essentially zero at some point in the flow,

then the stagnation conditions will be those existing at

the zero velocity point.


COMPRESSIBLE FLOW

However, even when the flow is non-isentropic, the

concept of the stagnation conditions is still useful, the

stagnation conditions at a point the being the conditions

that would exist if the local flow were brought to rest

isentropically.

If the equations derived in the previous section are

applied between a point in the flow where the pressure,

density, temperature and Mach number are P, , T, M

respectively, then if the stagnation conditions are

denoted by the subscript 0, the stagnation pressure,

density and temperature will, since the Mach number is

zero at the point where the stagnation conditions exist,

be given by ;


COMPRESSIBLE FLOW

120

211

−−+=

γγ

γ MPP

11

20

211

−−+=

γγρρ M

−+= 20

211 M

TT γ

( for the particular case of 4.1=γ )


COMPRESSIBLE FLOW

CRITICAL CONDITIONS

The critical conditions are those that would exist if the

flow was isentropically accelerated or decelerated until

the Mach number was unity, (M = 1)

These critical conditions are usually denoted by an

asterisk.

By setting M2=1, we found ;

+−

++

= 2*

11

12 M

TT

γγ

γ

21

2*

11

12

+−

++

= Maa

γγ

γ

12

*

11

12 −

+−

++

=γγ

γγ

γM

PP


COMPRESSIBLE FLOW

11

2*

11

12 −

+−

++

=γ

γγ

γρρ M

By setting M2=0, we found ;

12

0

*

+=γT

T

12

0

*

+=

γaa

1

0

*

12 −

+=

γγ

γPP

11

0

*

12 −

+=

γ

γρρ

For the case of air flow ;

833.00

*

=TT

, 528.00

*

=PP

, 634.00

*

=ρρ


COMPRESSIBLE FLOW

MAXIMUM DISCHARGE VELOCITY

Also known as “maximum escape velocity”, is the

velocity that would be generated if a gas was

adiabatically expanded until its temperature has

dropped to absolute zero.

Using the adiabatic energy equation gives the maximum

discharge velocity as :

0

22

22

ˆTcTcVVpP =+=

This can be rearranged to give ;

02 2)2(ˆ TcTcVV pP =+=

12)

12(ˆ

20

22

−=

−+=

γγaaVV


ONE-DIMENSIONAL ISENTROPIC FLOW SUMMARY OF MAJOR EQUATIONS ISENTROPIC FLOW RELATIONS

γ

ρρ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

2

1

2

PP

γγγ

ρρ

1

1

2

1

1

2

1

2

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

PP

TT

γγ

γγ

ρρ 2

1

1

2

1

1

221

1

2

1

2

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

PP

TT

aa

1

222

1

212

1

1

2

)1(1)1(1 −

⎥⎦

⎤⎢⎣

⎡

−+−+

=γγ

γγ

MM

PP

222

1

212

1

1

2

)1(1)1(1MM

TT

−+−+

=γγ

11

222

1

212

1

1

2

)1(1)1(1 −

⎥⎦

⎤⎢⎣

⎡

−+−+

=γ

γγ

ρρ

MM

)1(21

212

1

222

1

2

1

1

2

)1(1)1(1 −

+

⎥⎦

⎤⎢⎣

⎡−+−+

=γγ

γγ

MM

MM

AA

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

2

1

1

2

1

2

AA

VV

ρρ

STAGNATION CONDITIONS

12210 )1(1 −−+= γ

γ

γ MPP

2210 )1(1 M

TT

−+= γ

11

2210 )1(1 −−+= γγ

ρρ M

CRITICAL CONDITIONS

12

*

11

12 −

⎥⎦

⎤⎢⎣

⎡+−

++

=γγ

γγ

γM

PP

2*

11

12 M

TT

+−

++

=γγ

γ

11

2*

11

12 −

⎥⎦

⎤⎢⎣

⎡+−

++

=γ

γγ

γρρ M

RELATIONSHIP BETWEEN CRITICAL AND STAGNATION CONDITIONS

1

0

*

12 −

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=γγ

γPP

12

0

*

+=γT

T

11

0

*

12 −

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=γ

γρρ

12

0

*

+=

γaa

Table 1

Some typical values for the speed of sound at 0 ºC

Source:

Compressible Fluid Flow , The McGraw-Hill Companies, Inc.

Patrick, H.O. and William, E.C.

Speed of sound

at 0 ºC (m/s)

Air 289.66 1.404 331

Argon (Ar) 39.94 1.667 308

Carbon dioxide (CO2) 44.01 1.3 258

Freon 12 (CCl2F2) 120.9 1.139 146

Helium (He) 4.003 1.667 970

Hydrogen (H2) 2.016 1.407 1270

Xenon (Xe) 131.3 1.667 170

γMolar massGas

Table 2

Isentropic flow tables for air with γ = 1.4

Source:



M T o / T P o / P ρ o / ρ a o / a A / A* θ0.00 1.00000 1.00000 1.00000 1.00000 - -0.02 1.00008 1.00028 1.00020 1.00004 28.94213 -0.04 1.00032 1.00112 1.00080 1.00016 14.48149 -0.06 1.00072 1.00252 1.00180 1.00036 9.66591 -0.08 1.00128 1.00449 1.00320 1.00064 7.26161 -0.10 1.00200 1.00702 1.00501 1.00100 5.82183 -0.12 1.00288 1.01012 1.00722 1.00144 4.86432 -0.14 1.00392 1.01379 1.00983 1.00196 4.18240 -0.16 1.00512 1.01803 1.01285 1.00256 3.67274 -0.18 1.00648 1.02286 1.01628 1.00323 3.27793 -0.20 1.00800 1.02828 1.02012 1.00399 2.96352 -0.22 1.00968 1.03429 1.02438 1.00483 2.70760 -0.24 1.01152 1.04090 1.02905 1.00574 2.49556 -0.26 1.01352 1.04813 1.03414 1.00674 2.31729 -0.28 1.01568 1.05596 1.03966 1.00781 2.16555 -0.30 1.01800 1.06443 1.04561 1.00896 2.03507 -0.32 1.02048 1.07353 1.05199 1.01019 1.92185 -0.34 1.02312 1.08329 1.05881 1.01149 1.82288 -0.36 1.02592 1.09370 1.06607 1.01288 1.73578 -0.38 1.02888 1.10478 1.07377 1.01434 1.65870 -0.40 1.03200 1.11655 1.08193 1.01587 1.59014 -0.42 1.03528 1.12902 1.09055 1.01749 1.52890 -0.44 1.03872 1.14221 1.09963 1.01918 1.47401 -0.46 1.04232 1.15612 1.10918 1.02094 1.42463 -0.48 1.04608 1.17078 1.11921 1.02278 1.38010 -0.50 1.05000 1.18621 1.12973 1.02470 1.33984 -0.52 1.05408 1.20242 1.14073 1.02668 1.30339 -0.54 1.05832 1.21944 1.15224 1.02875 1.27032 -

M T o / T P o / P ρ o / ρ a o / a A / A* θ0.56 1.06272 1.23727 1.16425 1.03088 1.24029 -0.58 1.06728 1.25596 1.17678 1.03309 1.21301 -0.60 1.07200 1.27550 1.18984 1.03537 1.18820 -0.62 1.07688 1.29594 1.20342 1.03773 1.16565 -0.64 1.08192 1.31729 1.21755 1.04015 1.14515 -0.66 1.08712 1.33959 1.23224 1.04265 1.12654 -0.68 1.09248 1.36285 1.24748 1.04522 1.10965 -0.70 1.09800 1.38710 1.26330 1.04785 1.09437 -0.72 1.10368 1.41238 1.27970 1.05056 1.08057 -0.74 1.10952 1.43871 1.29670 1.05334 1.06814 -0.76 1.11552 1.46612 1.31430 1.05618 1.05700 -0.78 1.12168 1.49466 1.33252 1.05909 1.04705 -0.80 1.12800 1.52434 1.35137 1.06207 1.03823 -0.82 1.13448 1.55521 1.37086 1.06512 1.03046 -0.84 1.14112 1.58730 1.39100 1.06823 1.02370 -0.86 1.14792 1.62066 1.41182 1.07141 1.01787 -0.88 1.15488 1.65531 1.43332 1.07465 1.01294 -0.90 1.16200 1.69130 1.45551 1.07796 1.00886 -0.92 1.16928 1.72868 1.47841 1.08133 1.00560 -0.94 1.17672 1.76749 1.50204 1.08477 1.00311 -0.96 1.18432 1.80776 1.52642 1.08826 1.00136 -0.98 1.19208 1.84956 1.55154 1.09182 1.00034 -1.00 1.20000 1.89293 1.57744 1.09545 1.00000 -1.02 1.20808 1.93792 1.60413 1.09913 1.00033 0.125681.04 1.21632 1.98457 1.63162 1.10287 1.00131 0.350971.06 1.22472 2.03296 1.65994 1.10667 1.00291 0.636681.08 1.23328 2.08313 1.68910 1.11053 1.00512 0.968031.10 1.24200 2.13514 1.71911 1.11445 1.00793 1.336191.12 1.25088 2.18905 1.75000 1.11843 1.01131 1.735031.14 1.25992 2.24492 1.78179 1.12246 1.01527 2.159941.16 1.26912 2.30282 1.81450 1.12655 1.01978 2.607331.18 1.27848 2.36281 1.84814 1.13070 1.02484 3.074241.20 1.28800 2.42497 1.88274 1.13490 1.03044 3.558221.22 1.29768 2.48935 1.91831 1.13916 1.03657 4.057181.24 1.30752 2.55605 1.95488 1.14347 1.04323 4.569341.26 1.31752 2.62513 1.99248 1.14783 1.05041 5.093131.28 1.32768 2.69666 2.03111 1.15225 1.05810 5.627171.30 1.33800 2.77074 2.07081 1.15672 1.06630 6.170261.32 1.34848 2.84745 2.11160 1.16124 1.07502 6.721311.34 1.35912 2.92686 2.15350 1.16581 1.08424 7.279341.36 1.36992 3.00908 2.19653 1.17044 1.09396 7.843481.38 1.38088 3.09418 2.24073 1.17511 1.10419 8.412941.40 1.39200 3.18227 2.28612 1.17983 1.11493 8.987001.42 1.40328 3.27345 2.33271 1.18460 1.12616 9.564991.44 1.41472 3.36780 2.38054 1.18942 1.13790 10.146331.46 1.42632 3.46545 2.42964 1.19429 1.15015 10.73047

M T o / T P o / P ρ o / ρ a o / a A / A* θ1.48 1.43808 3.56649 2.48003 1.19920 1.16290 11.316911.50 1.45000 3.67103 2.53175 1.20416 1.17617 11.905181.52 1.46208 3.77919 2.58481 1.20917 1.18994 12.494861.54 1.47432 3.89109 2.63924 1.21422 1.20423 13.085571.56 1.48672 4.00684 2.69509 1.21931 1.21904 13.676931.58 1.49928 4.12657 2.75237 1.22445 1.23438 14.268621.60 1.51200 4.25041 2.81112 1.22963 1.25024 14.860321.62 1.52488 4.37849 2.87137 1.23486 1.26663 15.451771.64 1.53792 4.51095 2.93315 1.24013 1.28355 16.042681.66 1.55112 4.64792 2.99649 1.24544 1.30102 16.632821.68 1.56448 4.78955 3.06143 1.25079 1.31904 17.221951.70 1.57800 4.93599 3.12801 1.25618 1.33761 17.809881.72 1.59168 5.08739 3.19624 1.26162 1.35674 18.396401.74 1.60552 5.24391 3.26617 1.26709 1.37643 18.981341.76 1.61952 5.40570 3.33784 1.27260 1.39670 19.564531.78 1.63368 5.57294 3.41128 1.27815 1.41755 20.145801.80 1.64800 5.74580 3.48653 1.28374 1.43898 20.725031.82 1.66248 5.92444 3.56362 1.28937 1.46101 21.302081.84 1.67712 6.10906 3.64259 1.29504 1.48365 21.876821.86 1.69192 6.29984 3.72348 1.30074 1.50689 22.449141.88 1.70688 6.49696 3.80634 1.30648 1.53076 23.018931.90 1.72200 6.70064 3.89119 1.31225 1.55526 23.586101.92 1.73728 6.91106 3.97809 1.31806 1.58039 24.150561.94 1.75272 7.12843 4.06707 1.32390 1.60617 24.712231.96 1.76832 7.35297 4.15817 1.32978 1.63261 25.271021.98 1.78408 7.58490 4.25144 1.33569 1.65972 25.826882.00 1.80000 7.82445 4.34692 1.34164 1.68750 26.37973

Table 3

Approximate properties of the standard atmosphere

Source:



H T P ρ a v( m ) ( K ) ( Pa × 105 ) ( kg/m3 ) ( m/s ) ( m2/s × 10-5 )

0 288.16 1.0133 1.2250 340.28 1.4610200 286.86 0.9895 1.2017 339.52 1.4841400 285.56 0.9662 1.1787 338.75 1.5077600 284.26 0.9433 1.1560 337.97 1.5318800 282.97 0.9209 1.1337 337.21 1.55641000 281.67 0.8989 1.1118 336.43 1.58141200 280.37 0.8773 1.0901 335.65 1.60701400 279.07 0.8561 1.0688 334.87 1.63311600 277.77 0.8354 1.0478 334.09 1.65981800 276.47 0.8151 1.0271 333.31 1.68702000 275.17 0.7952 1.0067 332.53 1.71482200 273.87 0.7756 0.9866 331.74 1.74312400 272.58 0.7565 0.9669 330.96 1.77212600 271.28 0.7377 0.9474 330.17 1.80172800 269.98 0.7194 0.9283 329.37 1.83193000 268.68 0.7014 0.9094 328.58 1.86273200 267.38 0.6837 0.8909 327.79 1.89433400 266.08 0.6665 0.8726 326.99 1.92653600 264.78 0.6495 0.8546 326.19 1.95943800 263.49 0.6330 0.8369 325.39 1.99304000 262.19 0.6167 0.8195 324.59 2.02734200 260.89 0.6008 0.8023 323.78 2.09834400 259.59 0.5855 0.7856 322.97 2.11674600 258.29 0.5701 0.7689 322.17 2.13504800 254.39 0.5552 0.7526 319.72 2.17255000 255.69 0.5406 0.7365 320.54 2.2109

H T P ρ a v( m ) ( K ) ( Pa × 105 ) ( kg/m3 ) ( m/s ) ( m2/s × 10-5 )5200 254.39 0.5263 0.7207 319.72 2.25015400 253.10 0.5123 0.7052 318.91 2.29025600 251.80 0.4987 0.6899 318.09 2.37325800 250.50 0.4855 0.6751 317.27 2.39476000 249.20 0.4722 0.6602 316.45 2.41616200 247.90 0.4594 0.6465 315.62 2.46006400 246.60 0.4464 0.6314 314.79 2.50496600 245.30 0.4347 0.6173 313.96 2.55096800 244.01 0.4227 0.6035 313.13 2.59807000 242.71 0.4110 0.5900 312.30 2.64627200 241.41 0.3996 0.5767 311.46 2.69557400 240.11 0.3884 0.5636 310.62 2.74607600 238.81 0.3775 0.5507 309.78 2.79777800 237.51 0.3668 0.5381 308.93 2.85068000 236.21 0.3564 0.5257 308.09 2.90488200 234.91 0.3462 0.5135 307.24 2.96048400 233.62 0.3363 0.5015 306.39 3.01738600 232.32 0.3266 0.4898 305.54 3.07568800 231.02 0.3171 0.4782 304.68 3.13539000 229.72 0.3079 0.4669 303.83 3.19669200 228.42 0.2988 0.4448 302.96 3.32379400 227.12 0.2901 0.4395 302.10 3.35679600 225.82 0.2814 0.4341 301.24 3.38969800 224.53 0.2730 0.4236 300.37 3.4573

10000 223.23 0.2648 0.4132 299.50 3.5266

Tutorial 1 for compressible flow

1. An air stream enters a variable area channel at a velocity of 30m/s with a pressure of 120kPa and a temperature of 10ºC. At a certain point in the channel, the velocity is found to be 250m/s. Using Bernoulli’s equation (i.e., p+ρV2/2=constant), which assunmes incompressible flow, find the pressure at this point. In this calculation use the density evaluated at the inlet conditions. If the temperature of the air is assumed to remain constant, evaluate the air density at the point in the flow where the velocity is 250m/s. Compare this density with the density a the inlet to the channel. On the basis of this comparison, do you think that the use of Bernoulli’s equation is justified.

2. Two kilograms of air at an initial temperature and pressure of 30ºC and 100kPa undergoes an isentropic process, the final temperature attained being 850ºC. Find the final pressure, the initial and final densities and the initial and final volumes.

3. Two air streams are mixed in a chamber. One stream enters the chamber through a 5cm diameter pipe at velocity of 100m/s with a pressure of 150kPa and a temperature of 30ºC. The other stream enters the chamber through a 1.5cm diameter pipe at a velocity of 150m/s with a pressure of 75kPa and a temperature of 30ºC. The air leaves the chamber through a 9cm diameter pipe at a pressure of 90kPa and a temperature of 30ºC. Assuming that the flow is steady, find the velocity in the exit pipe.

4. The jet engine fitted to a small aircraft uses 35kg/s of air when the aircraft is flying at a speed of 800km/h. The jet efflux velocity is 590m/s. If the pressure on the engine discharge plane is assumed to be equal to the ambient pressure and if effects of the mass of the fuel used are ignored, find the thrust developed by the engine.

5. The engine of a small jet aircraft develops a thrust of 18kN when the aircraft is flying at a speed of 900km/h at an altitude where the ambient pressure is 50kPa. The air flow rate through the engine is 75kg/s and the engine uses fuel at a rate of 3kg/s. The pressure on the engine discharge plane is 55kPa and the area of the engine exit is 0.2m2. Find the jet efflux velocity.

6. A solid fuelled rocket is fitted with a convergent-divergent nozzle with an exit plane diameter of 30cm. The pressure and velocity on this nozzle exit plane are 75Kpa and 750m/s respectively and the mass flow rate through the nozzle is 350kg/s. Find the thrust developed by this engine when the ambient pressure is

(a) 100kPa and (b) 20kPa.

7. In a hydrogen powered rocket, hydrogen enters a nozzle at a very low velocity with a temperature and pressure of 2000ºC and 6.8Mpa respectively. The pressure on the exit plane of the nozzle is equal to the ambient pressure which is 10kPa. If the required thrust is 10MN, what hydrogen mass flow rate required? The flow though the nozzle can be assumed to be isentropic and the specific heat ratio of the hydrogen can be assumed to be 1.4.

8. Carbon dioxide flows through a constant area duct. At inlet to the duct, the velocity is 120m/s and the temperature and pressure are 200ºC and 700kPa respectively. Heat is added to the flow in the duct and at the exit of the duct the velocity is 240m/s and the temperature is 450ºC. Find the amount of heat being added to the carbon dioxide per unit mass of gas and the mass flow rate through the duct per unit cross-sectional area of the duct. Assume that for carbon dioxide, γ=1.3.

9. Air enters a heat exchanger with a velocity of 120m/s and a temperature and pressure of 225ºC and 2.5Mpa. Heat is removed from the air in the heat exchanger and the air leaves with a velocity 30m/s at a temperature and pressure 80ºC and 2.45MPa. Find the heat removed per kilogram of air flowing through the heat exchanger and the density of the air at the inlet and the exit to the heat exchanger.


1. Air enters a tank at a velocity of 100m/s and leaves the tank at a velocity of 200m/s. lf the flow is adiabatic find the difference between the temperature of the air at exit and the temperature of the air at inlet･

2. Air at a temperature of 25°C is flowing at a velocity of 500m/s. A shock wave

occurs in the flow reducing the velocity to 300m/s. Assuming the flow through the shock wave to be adiabatic, find the temperature of the air behind the shock wave.

3. Air being released from a tire through the valve is found to have a temperature

of 15°C. Assuming that the air in the tire is at the ambient temperature of 30°C, find the velocity of the air at the exit of the valve. The process can be assumed to be adiabatic.

4. Gas with a molecular weight of 4 and a specific heat ratio of 1.67 flows

through a variable area duct. At some point in the now the velocity is 180m/s and the temperature is 10°C. At some other point in the flow the temperature is minus 10°C. Find the velocity at this point in the flow assuming that the flow is adiabatic.

5. At a section of a circular duct through which air is flowing the pressure is

150kPa, the temperature is 35°C, the velocity is 250m/s, and the diameter is 0.2m. If, at this section， the duct diameter is increasing at a rate of 0.1m/m, find dp/dx, dV/dx and dρ/dx

6. Consider an isothermal air flow through a duct. At a certain section of the duct

the velocity， temperature，and pressure are 200m/s, 25°C,and 120kPa respectively. lf the velocity is decreasing at this section at a rate of 30 percent per meter, find dp/dx, ds/dx and dρ/dx.

7. Consider adiabatic air flow through a variable area duct, At a certain section of

the duct the flow area is 0.1m2, the pressure is 120kPa, the temperature is 15°C and the duct area is changing at a rate of 0.1m2/m. Plot the variations of dp/dx, dV/dx and dρ/dx with the velocity at the section for velocities between 50m/s and 300m/s.


1. The velocity of an air flow changes by one percent. Assuming that the flow is isentropic，plot the percentage changes in pressure, temperature, and density induced by this change in velocity with flow Mach number for Mach numbers between 0.2 and 2.

2. Calculate the speed of sound at 288 K in hydrogen, helium and nitrogen. Under

what conditions will the speed of sound in hydrogen be equal to that in helium?

3. Find the speed of sound in carbon dioxide at temperatures of 20ºC and 600ºC.

4. A very weak pressure wave, i.e, a sound wave, across which the pressure rise is 30Pa moves through air which has a temperature of 30°C and a pressure of 101kPa. Find the density change, the temperature change, and the velocity change across this wave.

5. An airplane can fly at a speed or800km/h at sea-level where the temperature is

15°C. lf the airplane flies at the same Mach number at an altitude where the temperature is -44°C, find the speed at which the airplane is flying at this altitude.

6. The test section of a supersonic wind tunnel is square in cross-section with a side

length of 1.22m. The Mach number in the test section is 3.5, the temperature is -100°C, and the pressure is 20kPa. Find the mass flow rate of air through the test section.

7. A gas with a molar mass of 44 and a specific heat ratio 1.67 flows through a

channel at supersonic speed. The temperature of the gas in the channel is 10°C. A photograph of the flow reveals weak waves originating at imperfections in the wall running across the flow at an angle to 45° to the flow direction. Find the Mach number and the velocity in the flow.

8. An observer at sea level does not hear an aircraft that is flying at an altitude of

7000m until it is a distance of 13km from the observer. Estimate the Mach number at which the aircraft is flying. In arriving at the answer, assume that the average temperature of the air between sea level and 7000m is -10°C.

9. An observer on the ground finds that an airplane flying horizontally at an altitude

of 2500m has traveled 6 km from the overhead position before the sound of the airplane is first heard. Assuming that, overall, the aircraft creates a small disturbance, estimate the speed at which the airplane is flying. The average air temperature between the ground and the altitude at which the airplane is flying is 10°C. Explain the assumptions you have made in arriving at the answer.


1. A gas with a molar mass of 4 and a specific heat ratio of 1.67 flows through a variable area duct. At some point in the flow the velocity is 200m/s and the temperature is 10ºC. Find the Mach number at this point in the flow. At some other point in the flow the temperature is -10ºC. Find the velocity and Mach number at this point in the flow assuming that the flow is isentropic.

2. The exhaust gases from a rocket engine have a molar mass of 14. They can be

assumed to behave as a perfect gas with a specific heat ratio of 1.25. These gases are accelerated through a nozzle. At some point in the nozzle where the cross-sectional area of the nozzle is 0.7m2, the pressure is 1000kPa, the temperature is 500°C and the velocity is 100 m/s, Find the mass flow rate through the nozzle and the stagnation pressure and temperature. Also find the highest velocity that could be generated by expanding this flow. lf the pressure at some other point in the nozzle is 100kPa, find the temperature and velocity at this point in the flow assuming the flow to be one-dimensional and isentropic.

3. lf Concorde is flying at a Mach number of 2.2, at an altitude of 10000m in the

standard atmosphere, find the stagnation pressure and temperature for the flow over the aircraft.

4. If a gas is flowing at 300m/s and has a pressure and temperature of 90kPa and

20°C, find the maximum possible velocity that could be generated by expansion of this gas if the gas is air and if it is helium.

\

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